WORST_CASE(Omega(n^1),O(n^1))
proof of input_8J57vUKyYR.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTRS
    (11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTRS
    (13) CpxTrsMatchBoundsTAProof [FINISHED, 18 ms]
    (14) BOUNDS(1, n^1)
(15) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
    (17) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 4 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 295 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
            (32) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
S tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, IF_3, G_2

Compound Symbols:   c1, c2, c3_1, c4, c5, c6_1, c7_3, c8_3


----------------------------------------

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   IF(true, s(z0), s(z1)) -> c4
   F(1) -> c2
   F(0) -> c1
   IF(false, s(z0), s(z1)) -> c5

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_3, c8_3


----------------------------------------

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_1, c8_1


----------------------------------------

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_1, c8_1


----------------------------------------

(9) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
s0(0) -> 0
c30(0) -> 0
c0(0) -> 0
c60(0) -> 0
c70(0) -> 0
c80(0) -> 0
F0(0) -> 1
G0(0, 0) -> 2
F1(0) -> 3
c31(3) -> 1
G1(0, 0) -> 4
c61(4) -> 2
F1(0) -> 5
c71(5) -> 2
s1(0) -> 7
G1(7, 0) -> 6
c81(6) -> 2
c31(3) -> 3
c31(3) -> 5
c61(4) -> 4
G1(7, 0) -> 4
c61(4) -> 6
c71(5) -> 4
F1(7) -> 5
c71(5) -> 6
c81(6) -> 4
s1(7) -> 7
c81(6) -> 6
F2(0) -> 8
c32(8) -> 5
F2(7) -> 8
c31(3) -> 8
c32(8) -> 8

----------------------------------------

(14)
BOUNDS(1, n^1)

----------------------------------------

(15) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
S tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, IF_3, G_2

Compound Symbols:   c1, c2, c3_1, c4, c5, c6_1, c7_3, c8_3


----------------------------------------

(17) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))

The (relative) TRS S consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Rewrite Strategy: INNERMOST
----------------------------------------

(19) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   F(0') -> c1
   F(1') -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))

The (relative) TRS S consists of the following rules:

   f(0') -> true
   f(1') -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Rewrite Strategy: INNERMOST
----------------------------------------

(21) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
F(0') -> c1
F(1') -> c2
F(s(z0)) -> c3(F(z0))
IF(true, s(z0), s(z1)) -> c4
IF(false, s(z0), s(z1)) -> c5
G(z0, c(z1)) -> c6(G(z0, z1))
G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
f(0') -> true
f(1') -> false
f(s(z0)) -> f(z0)
if(true, s(z0), s(z1)) -> s(z0)
if(false, s(z0), s(z1)) -> s(z1)
g(z0, c(z1)) -> c(g(z0, z1))
g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Types:
F :: 0':1':s:c -> c1:c2:c3
0' :: 0':1':s:c
c1 :: c1:c2:c3
1' :: 0':1':s:c
c2 :: c1:c2:c3
s :: 0':1':s:c -> 0':1':s:c
c3 :: c1:c2:c3 -> c1:c2:c3
IF :: true:false -> 0':1':s:c -> 0':1':s:c -> c4:c5
true :: true:false
c4 :: c4:c5
false :: true:false
c5 :: c4:c5
G :: 0':1':s:c -> 0':1':s:c -> c6:c7:c8
c :: 0':1':s:c -> 0':1':s:c
c6 :: c6:c7:c8 -> c6:c7:c8
c7 :: c6:c7:c8 -> c4:c5 -> c1:c2:c3 -> c6:c7:c8
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
f :: 0':1':s:c -> true:false
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c8 :: c6:c7:c8 -> c4:c5 -> c6:c7:c8 -> c6:c7:c8
hole_c1:c2:c31_9 :: c1:c2:c3
hole_0':1':s:c2_9 :: 0':1':s:c
hole_c4:c53_9 :: c4:c5
hole_true:false4_9 :: true:false
hole_c6:c7:c85_9 :: c6:c7:c8
gen_c1:c2:c36_9 :: Nat -> c1:c2:c3
gen_0':1':s:c7_9 :: Nat -> 0':1':s:c
gen_c6:c7:c88_9 :: Nat -> c6:c7:c8

----------------------------------------

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, G, f, g

They will be analysed ascendingly in the following order:
F < G
f < G
g < G
f < g

----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
F(0') -> c1
F(1') -> c2
F(s(z0)) -> c3(F(z0))
IF(true, s(z0), s(z1)) -> c4
IF(false, s(z0), s(z1)) -> c5
G(z0, c(z1)) -> c6(G(z0, z1))
G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
f(0') -> true
f(1') -> false
f(s(z0)) -> f(z0)
if(true, s(z0), s(z1)) -> s(z0)
if(false, s(z0), s(z1)) -> s(z1)
g(z0, c(z1)) -> c(g(z0, z1))
g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Types:
F :: 0':1':s:c -> c1:c2:c3
0' :: 0':1':s:c
c1 :: c1:c2:c3
1' :: 0':1':s:c
c2 :: c1:c2:c3
s :: 0':1':s:c -> 0':1':s:c
c3 :: c1:c2:c3 -> c1:c2:c3
IF :: true:false -> 0':1':s:c -> 0':1':s:c -> c4:c5
true :: true:false
c4 :: c4:c5
false :: true:false
c5 :: c4:c5
G :: 0':1':s:c -> 0':1':s:c -> c6:c7:c8
c :: 0':1':s:c -> 0':1':s:c
c6 :: c6:c7:c8 -> c6:c7:c8
c7 :: c6:c7:c8 -> c4:c5 -> c1:c2:c3 -> c6:c7:c8
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
f :: 0':1':s:c -> true:false
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c8 :: c6:c7:c8 -> c4:c5 -> c6:c7:c8 -> c6:c7:c8
hole_c1:c2:c31_9 :: c1:c2:c3
hole_0':1':s:c2_9 :: 0':1':s:c
hole_c4:c53_9 :: c4:c5
hole_true:false4_9 :: true:false
hole_c6:c7:c85_9 :: c6:c7:c8
gen_c1:c2:c36_9 :: Nat -> c1:c2:c3
gen_0':1':s:c7_9 :: Nat -> 0':1':s:c
gen_c6:c7:c88_9 :: Nat -> c6:c7:c8


Generator Equations:
gen_c1:c2:c36_9(0) <=> c1
gen_c1:c2:c36_9(+(x, 1)) <=> c3(gen_c1:c2:c36_9(x))
gen_0':1':s:c7_9(0) <=> 0'
gen_0':1':s:c7_9(+(x, 1)) <=> s(gen_0':1':s:c7_9(x))
gen_c6:c7:c88_9(0) <=> hole_c6:c7:c85_9
gen_c6:c7:c88_9(+(x, 1)) <=> c6(gen_c6:c7:c88_9(x))


The following defined symbols remain to be analysed:
F, G, f, g

They will be analysed ascendingly in the following order:
F < G
f < G
g < G
f < g

----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
F(gen_0':1':s:c7_9(n10_9)) -> gen_c1:c2:c36_9(n10_9), rt in Omega(1 + n10_9)

Induction Base:
F(gen_0':1':s:c7_9(0)) ->_R^Omega(1)
c1

Induction Step:
F(gen_0':1':s:c7_9(+(n10_9, 1))) ->_R^Omega(1)
c3(F(gen_0':1':s:c7_9(n10_9))) ->_IH
c3(gen_c1:c2:c36_9(c11_9))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Complex Obligation (BEST)

----------------------------------------

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(0') -> c1
F(1') -> c2
F(s(z0)) -> c3(F(z0))
IF(true, s(z0), s(z1)) -> c4
IF(false, s(z0), s(z1)) -> c5
G(z0, c(z1)) -> c6(G(z0, z1))
G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
f(0') -> true
f(1') -> false
f(s(z0)) -> f(z0)
if(true, s(z0), s(z1)) -> s(z0)
if(false, s(z0), s(z1)) -> s(z1)
g(z0, c(z1)) -> c(g(z0, z1))
g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Types:
F :: 0':1':s:c -> c1:c2:c3
0' :: 0':1':s:c
c1 :: c1:c2:c3
1' :: 0':1':s:c
c2 :: c1:c2:c3
s :: 0':1':s:c -> 0':1':s:c
c3 :: c1:c2:c3 -> c1:c2:c3
IF :: true:false -> 0':1':s:c -> 0':1':s:c -> c4:c5
true :: true:false
c4 :: c4:c5
false :: true:false
c5 :: c4:c5
G :: 0':1':s:c -> 0':1':s:c -> c6:c7:c8
c :: 0':1':s:c -> 0':1':s:c
c6 :: c6:c7:c8 -> c6:c7:c8
c7 :: c6:c7:c8 -> c4:c5 -> c1:c2:c3 -> c6:c7:c8
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
f :: 0':1':s:c -> true:false
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c8 :: c6:c7:c8 -> c4:c5 -> c6:c7:c8 -> c6:c7:c8
hole_c1:c2:c31_9 :: c1:c2:c3
hole_0':1':s:c2_9 :: 0':1':s:c
hole_c4:c53_9 :: c4:c5
hole_true:false4_9 :: true:false
hole_c6:c7:c85_9 :: c6:c7:c8
gen_c1:c2:c36_9 :: Nat -> c1:c2:c3
gen_0':1':s:c7_9 :: Nat -> 0':1':s:c
gen_c6:c7:c88_9 :: Nat -> c6:c7:c8


Generator Equations:
gen_c1:c2:c36_9(0) <=> c1
gen_c1:c2:c36_9(+(x, 1)) <=> c3(gen_c1:c2:c36_9(x))
gen_0':1':s:c7_9(0) <=> 0'
gen_0':1':s:c7_9(+(x, 1)) <=> s(gen_0':1':s:c7_9(x))
gen_c6:c7:c88_9(0) <=> hole_c6:c7:c85_9
gen_c6:c7:c88_9(+(x, 1)) <=> c6(gen_c6:c7:c88_9(x))


The following defined symbols remain to be analysed:
F, G, f, g

They will be analysed ascendingly in the following order:
F < G
f < G
g < G
f < g

----------------------------------------

(28) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(29)
BOUNDS(n^1, INF)

----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
F(0') -> c1
F(1') -> c2
F(s(z0)) -> c3(F(z0))
IF(true, s(z0), s(z1)) -> c4
IF(false, s(z0), s(z1)) -> c5
G(z0, c(z1)) -> c6(G(z0, z1))
G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
f(0') -> true
f(1') -> false
f(s(z0)) -> f(z0)
if(true, s(z0), s(z1)) -> s(z0)
if(false, s(z0), s(z1)) -> s(z1)
g(z0, c(z1)) -> c(g(z0, z1))
g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Types:
F :: 0':1':s:c -> c1:c2:c3
0' :: 0':1':s:c
c1 :: c1:c2:c3
1' :: 0':1':s:c
c2 :: c1:c2:c3
s :: 0':1':s:c -> 0':1':s:c
c3 :: c1:c2:c3 -> c1:c2:c3
IF :: true:false -> 0':1':s:c -> 0':1':s:c -> c4:c5
true :: true:false
c4 :: c4:c5
false :: true:false
c5 :: c4:c5
G :: 0':1':s:c -> 0':1':s:c -> c6:c7:c8
c :: 0':1':s:c -> 0':1':s:c
c6 :: c6:c7:c8 -> c6:c7:c8
c7 :: c6:c7:c8 -> c4:c5 -> c1:c2:c3 -> c6:c7:c8
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
f :: 0':1':s:c -> true:false
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c8 :: c6:c7:c8 -> c4:c5 -> c6:c7:c8 -> c6:c7:c8
hole_c1:c2:c31_9 :: c1:c2:c3
hole_0':1':s:c2_9 :: 0':1':s:c
hole_c4:c53_9 :: c4:c5
hole_true:false4_9 :: true:false
hole_c6:c7:c85_9 :: c6:c7:c8
gen_c1:c2:c36_9 :: Nat -> c1:c2:c3
gen_0':1':s:c7_9 :: Nat -> 0':1':s:c
gen_c6:c7:c88_9 :: Nat -> c6:c7:c8


Lemmas:
F(gen_0':1':s:c7_9(n10_9)) -> gen_c1:c2:c36_9(n10_9), rt in Omega(1 + n10_9)


Generator Equations:
gen_c1:c2:c36_9(0) <=> c1
gen_c1:c2:c36_9(+(x, 1)) <=> c3(gen_c1:c2:c36_9(x))
gen_0':1':s:c7_9(0) <=> 0'
gen_0':1':s:c7_9(+(x, 1)) <=> s(gen_0':1':s:c7_9(x))
gen_c6:c7:c88_9(0) <=> hole_c6:c7:c85_9
gen_c6:c7:c88_9(+(x, 1)) <=> c6(gen_c6:c7:c88_9(x))


The following defined symbols remain to be analysed:
f, G, g

They will be analysed ascendingly in the following order:
f < G
g < G
f < g

----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':1':s:c7_9(n286_9)) -> true, rt in Omega(0)

Induction Base:
f(gen_0':1':s:c7_9(0)) ->_R^Omega(0)
true

Induction Step:
f(gen_0':1':s:c7_9(+(n286_9, 1))) ->_R^Omega(0)
f(gen_0':1':s:c7_9(n286_9)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
F(0') -> c1
F(1') -> c2
F(s(z0)) -> c3(F(z0))
IF(true, s(z0), s(z1)) -> c4
IF(false, s(z0), s(z1)) -> c5
G(z0, c(z1)) -> c6(G(z0, z1))
G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
f(0') -> true
f(1') -> false
f(s(z0)) -> f(z0)
if(true, s(z0), s(z1)) -> s(z0)
if(false, s(z0), s(z1)) -> s(z1)
g(z0, c(z1)) -> c(g(z0, z1))
g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

Types:
F :: 0':1':s:c -> c1:c2:c3
0' :: 0':1':s:c
c1 :: c1:c2:c3
1' :: 0':1':s:c
c2 :: c1:c2:c3
s :: 0':1':s:c -> 0':1':s:c
c3 :: c1:c2:c3 -> c1:c2:c3
IF :: true:false -> 0':1':s:c -> 0':1':s:c -> c4:c5
true :: true:false
c4 :: c4:c5
false :: true:false
c5 :: c4:c5
G :: 0':1':s:c -> 0':1':s:c -> c6:c7:c8
c :: 0':1':s:c -> 0':1':s:c
c6 :: c6:c7:c8 -> c6:c7:c8
c7 :: c6:c7:c8 -> c4:c5 -> c1:c2:c3 -> c6:c7:c8
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
f :: 0':1':s:c -> true:false
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c8 :: c6:c7:c8 -> c4:c5 -> c6:c7:c8 -> c6:c7:c8
hole_c1:c2:c31_9 :: c1:c2:c3
hole_0':1':s:c2_9 :: 0':1':s:c
hole_c4:c53_9 :: c4:c5
hole_true:false4_9 :: true:false
hole_c6:c7:c85_9 :: c6:c7:c8
gen_c1:c2:c36_9 :: Nat -> c1:c2:c3
gen_0':1':s:c7_9 :: Nat -> 0':1':s:c
gen_c6:c7:c88_9 :: Nat -> c6:c7:c8


Lemmas:
F(gen_0':1':s:c7_9(n10_9)) -> gen_c1:c2:c36_9(n10_9), rt in Omega(1 + n10_9)
f(gen_0':1':s:c7_9(n286_9)) -> true, rt in Omega(0)


Generator Equations:
gen_c1:c2:c36_9(0) <=> c1
gen_c1:c2:c36_9(+(x, 1)) <=> c3(gen_c1:c2:c36_9(x))
gen_0':1':s:c7_9(0) <=> 0'
gen_0':1':s:c7_9(+(x, 1)) <=> s(gen_0':1':s:c7_9(x))
gen_c6:c7:c88_9(0) <=> hole_c6:c7:c85_9
gen_c6:c7:c88_9(+(x, 1)) <=> c6(gen_c6:c7:c88_9(x))


The following defined symbols remain to be analysed:
g, G

They will be analysed ascendingly in the following order:
g < G